What Is Discrete Distribution?
A discrete distribution is a probability distribution that depicts the occurrence of
discrete (individually countable) outcomes, such as 1, 2, 3... or zero vs. one. The
binomial distribution, for example, is a discrete distribution that evaluates the
probability of a "yes" or "no" outcome occurring over a given number of trials, given
the event's probability in each trial—such as flipping a coin one hundred times and
having the outcome be "heads".
Statistical distributions can be either discrete or continuous. A continuous
distribution is built from outcomes that fall on a continuum, such as all numbers
greater than 0 (which would include numbers whose decimals continue indefinitely,
such as pi = 3.14159265...). Overall, the concepts of discrete and continuous
probability distributions and the random variables they describe are the
underpinnings of probability theory and statistical analysis.
What Are the Two Requirements for a Discrete Probability Distribution?
The probabilities of random variables must have discrete (as opposed to continuous)
values as outcomes. For a cumulative distribution, the probability of each discrete
observation must be between 0 and 1; and the sum of the probabilities must equal
one (100%).
How Do You Know If a Distribution Is Discrete?
If there are only a set array of possible outcomes (e.g. only zero or one, or only
integers), then the data are discrete.
What Is the Binomial Distribution?
The binomial distribution is a probability distribution that summarizes the likelihood
that a value will take one of two independent values under a given set of parameters
or assumptions.
The underlying assumptions of the binomial distribution are that there is only one
outcome for each trial, that each trial has the same probability of success, and that
each trial is mutually exclusive, or independent of one another.
Analyzing Binomial Distribution
The expected value, or mean, of a binomial distribution, is calculated by multiplying
the number of trials (n) by the probability of successes (p), or n x p.
For example, the expected value of the number of heads in 100 trials of head and
tales is 50, or (100 * 0.5). Another common example of the binomial distribution is
by estimating the chances of success for a free-throw shooter in basketball where 1
= a basket is made and 0 = a miss.
The binomial distribution formula is calculated as:
P(x:n,p) = nCx x px(1-p)n-x
where:
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n is the number of trials (occurrences)
X is the number of successful trials
p is probability of success in a single trial
nCx is the combination of n and x. A combination is the number of ways to
choose a sample of x elements from a set of n distinct objects where order
does not matter and replacements are not allowed. Note that
nCx=n!/(r!(n−r)!), where ! is factorial (so, 4! = 4 x 3 x 2 x 1)
The mean of the binomial distribution is np, and the variance of the binomial
distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the
mean. When p > 0.5, the distribution is skewed to the left. When p < 0.5, the
distribution is skewed to the right.